{
  "id": "1401.4556",
  "version": "v4",
  "published": "2014-01-18T16:03:31.000Z",
  "updated": "2014-03-13T14:37:55.000Z",
  "title": "Kloosterman Sums with Multiplicative Coefficients",
  "authors": [
    "Ke Gong",
    "Chaohua Jia"
  ],
  "comment": "In this version we make some refinement",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f(n)$ be a multiplicative function satisfying $|f(n)|\\leq 1$, $q$ $(\\leq N^2)$ be a positive integer and $a$ be an integer with $(a,\\,q)=1$. In this paper, we shall prove that $$\\sum_{\\substack{n\\leq N\\\\ (n,\\,q)=1}}f(n)e({a\\bar{n}\\over q})\\ll\\sqrt{\\tau(q)\\over q}N\\log\\log(6N)+q^{{1\\over 4}+{\\epsilon\\over 2}}N^{1\\over 2}(\\log(6N))^{1\\over 2}+{N\\over \\sqrt{\\log\\log(6N)}},$$ where $\\bar{n}$ is the multiplicative inverse of $n$ such that $\\bar{n}n\\equiv 1\\,({\\rm mod}\\,q),\\,e(x)=\\exp(2\\pi ix),\\,\\tau(q)$ is the divisor function.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-03-13T14:37:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kloosterman sums",
      "multiplicative coefficients",
      "divisor function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.4556G"
    }
  }
}